Let #l# be a line described by equation ax+by+c=0 and let #P(x,y)# be a point not on #l#. Express the distance, #d# between #l and P# in terms of the coefficients #a, b and c# of the equation of line?

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Answer 1 · 2016-11-14T19:04:58

#d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)#

Let #l->a x + b y + c=0# and #p_0 = (x_0,y_0)# a point not on #l#.

Supposing that #b ne 0# and calling #d^2=(x-x_0)^2+(y-y_0)^2# after substituting #y=-(a x+c)/b# into #d^2# we have

#d^2=(x - x_0)^2 + ((c + a x)/b + y_0)^2#. The next step is find the #d^2# minimum regarding #x# so we will find #x# such that

#d/(dx)(d^2) = 2 (x - x_0) - (2 a ((c + a x)/b + y_0))/b = 0#. This occours for

#x = (b^2 x_0 - a b y_0-a c)/(a^2 + b^2)# Now, substituting this value into #d^2# we obtain

#d^2=(c + a x_0 + b y_0)^2/(a^2 + b^2)# so

#d = (c + a x_0 + b y_0)/sqrt(a^2 + b^2)#